A Self-Consistent Gibbs Excess Mixing Rule for Cubic Equations of State: derivation and fugacity coefficients

Transcription

1 A Self-Consstent Gbbs Excess Mxng Rule for Cubc Equatons of State: dervaton and fugacty coeffcents Paula B. Staudt, Rafael de P. Soares Departamento de Engenhara Químca, Escola de Engenhara, Unversdade Federal do Ro Grande do Sul, Rua Engenhero Lus Englert, s/n, Barro Farrouplha, CEP , Porto Alegre, RS, Brazl Abstract The extenson of the applcablty of cubc equatons of state (EoS) wth Gbbs excess models to the predcton of hgh-pressure/hgh-temperature vapor lqud equlbra of polar and/or asymmetrc s well known. In a recent work ( 1016/j.flud ) we have proposed the so called Self Consstent Mxng Rule (SCMR). The method was derved solely based on the assumpton of a zero excess volume lqud-lke phase. Tests wth substances dssmlar n sze, shape and chemcal nature have shown that any cubc equaton of state coupled wth the proposed mxng rule can reproduce the underlyng lqud actvty model at low pressures, showng that the method s self-consstent. Further, the method was extended for hgh pressures/temperatures by assumng a constant thermal expanson coeffcent lqud lke phase as the reference state. Very good results were obtaned when the proposed method was coupled wth Wlson, UNIQUAC, UNIFAC and a COSMO-based model n lqud lqud and vapor lqud equlbrum examples. The present document contans a mnmum descrpton of the SCMR method, ts dervaton and the equatons for the fugacty coeffcent. Key words: cubc equatons of state, EoS/G E mxng rules, Gbbs excess models, vapor-lqud equlbrum, lqud-lqud equlbrum. Correspondng author. Tel.: ; fax: Emal address: rafael@enq.ufrgs.br (Paula B. Staudt, Rafael de P. Soares) 1

2 1. Introducton In a recent work ( a new G E based mxng rule for cubc equatons of state was developed. The man advantage of the proposed method s that the combned model reproduces very well the Gγ E model t s based on wthout any addtonal emprcal correcton. For a detaled descrpton of the method as well as comparson wth smlar methods and expermental data, please refer to the full length manuscrpt avalable at dx.do.org/ /j.flud In ths document only the mxng rule dervaton and the fugacty coeffcent equatons are shown. 2. Mxng rule dervaton Most of the cubc equatons of state (EoS) avalable today are specal cases of a general cubc equaton [1], whch can be wrtten as: P = RT V b a (T) (V + ɛb) (V + σb) where P s the pressure, T s the temperature, V s the molar volume, ɛ and σ are constants for all substances and depend on the partcular EoS (see Table 1) and a (T) and b are, respectvely, the attractve and co-volume parameters specfc for each substance. The attractve a (T) and co-volume b parameters are usually determned usng generalzed correlatons based on crtcal propertes and acentrc factor, accordng to: a (T) = Ψ α (T r, ω) R 2 T 2 c P c (2) b = Ω RT c P c (3) where T c s the crtcal temperature, P c s the crtcal pressure, ω s the acentrc factor, T r = T/T c the reduced temperature and the other symbols are shown n Table 1. (1) 2.1. Mxng rule When dealng wth mxtures, the expressons for the attractve a and co-volume b parameters should be computed as a functon of the pure substances values a and b through mxng rules. 2

3 Table 1: Specfc cubc equaton parameters. EoS α(t r ) σ ɛ Ω Ψ van der Waals (vdw) /8 27/64 Redlch Kwong (RK) T 1/2 r Soave Redlch Kwong (SRK) α S RK (T r ; ω) a Peng-Robnson (PR) α PR (T r ; ω) b a α S RK (T r ; ω) = [ 1 + ( ω 0.176ω 2 ) ( 1 T r )] 2 b α PR (T r ; ω) = [ 1 + ( ω ω 2 ) ( 1 T r )] 2 The van der Waals (vdw) or classc mxng rule, present n most professonal process smulaton systems, s gven by: N a = x x j a a j (1 k j ) (4) =1 b = N x b (5) =1 where x s the mole fracton of the substance and k j s the bnary nteracton parameter, ntroduced to mprove the correlaton of phase equlbrum of mxtures. G E based mxng rules, n contrast to the classc mxng rule, obtan the nteracton nformaton from excess Gbbs energy Gγ E models, orgnally developed for the predcton of lqud actvty coeffcents γ. One possble expresson for computng the Gbbs excess energy from a cubc EoS s [2]: E RT = ln φ x ln φ (6) where φ s the mxture fugacty coeffcent and φ s the fugacty coeffcent of the pure substance, all n the same condtons of temperature and pressure. The fugacty coeffcent consdered n Equaton 6 for any cubc EoS n the generc form (Equaton 1) s gven by [3]: ln φ = (Z 1) ln(z β) + qi (7) 3

4 where Z PV/RT s the compressblty factor and the other auxlary varables are: β Pb/RT, q a/brt, I I 0 ln V+ɛb V+σb, and I 0 s a constant gven by 1/(σ ɛ). Usng Equaton 7 one can compute ln φ as well as ln φ by exchangng the mxture propertes (Z, β, q, and I) by the pure substance propertes (Z, β, q, and I ). Thus, by combnng Equaton 6 and Equaton 7, the Gbbs excess energy for a cubc EoS can be computed by: E RT = (Z 1) ln(z β) + qi x ((Z 1) ln(z β ) + q I ) (8) Usng the defnton of excess volume V E V x V and recallng that ln(z β) = ln P RT + ln(v b) one can obtan: E RT = PV E RT + x ln ( ) V b + qi V b x q I (9) The expresson gven by Equaton 9 contans no smplfcaton assumptons and can be used to get a fully consstent mxng rule f we make G E φ = GE γ. Although exact, the practcal use of Equaton 9 s lmted because t s an mplct mxng rule (the mxture volume V depends on q and vce versa). Now, by assumng that the excess volume s neglgble (V E = 0, V = V Id = x V ) the followng expresson s obtaned: E ( ) RT = V b x ln + qi Id x V Id q I (10) b where b should be computed wth the mxng rule Equaton 5, the volume of the pure substances V, as well as I Id and I, should be computed usng the lqud-lke root of the pure fluds at the system temperature and pressure. However, for cubc equatons, the attractve parameter a should depend on temperature and composton only. Snce most G E γ are developed for near atmospherc pressure, n ths work the lqud lke root requred for the determnaton of V s obtaned at 1 bar. The results would be essentally the same f a zero pressure s taken as reference. Fnally, by makng G E φ = GE γ a new explct mxng rule s obtaned by solatng q n Equaton 10: q = 1 G E ( ) γ I Id RT V b x ln + x V Id q I b (11) 4

5 The mxng rule gven by Equaton 11 wll reproduce the Gγ E model as long as the the system pressure s not too far from 1 bar and the zero excess volume assumpton holds. For all tests consdered (see the full-length manuscrpt) the E reproduced the Gγ E very well. Thus, the mxng rule gven by Equaton 11 was referred as the Self Consstent Mxng Rule (SCMR). The dervaton of fugacty coeffcents of substances n mxture accordng to the SCMR s gven n Secton Extenson for hgh pressure/temperature In the mxng rule proposed n the present work, the pure flud lqud lke volume of each substance n mxture s necessary. At hgh temperature condtons, usually above T r T/T c = 0.7, one can have problems wth fndng a lqud lke root from the cubc EoS. To crcumvent ths problem, an alternatve procedure s adopted n ths work to compute the pure flud lqud-lke molar volume to be used n Equaton 11. From the defnton of the volumetrc thermal expanson coeffcent of a pure flud β : β 1 ( ) V V T P and assumng a constant β, evaluated at a reference temperature T, the molar volume of a pure substance can be obtaned by the followng expresson: (12) ln V V = β (T T ) (13) The pure flud thermal expanson β, accordng to an EoS, s easly determned by ts defnton (Equaton 12). In ths work, the reference temperature T chosen was that to correspond to a T,r = 0.5. Ths temperature corresponds, approxmately, to the normal bolng temperature. Ths reference temperature assures a vald lqud lke root, consequently no problems wll occur to evaluate V. Then, n the SCMR mxng rule, the pure flud lqud lke root V s always evaluated by Equaton 13, allowng ts applcaton to hgh pressure/temperature and/or supercrtcal systems. 5

6 2.3. Polymer solutons Specally for polymer components, the requred pure component lqud-lke volume V = b /u was computed by consderng a constant and unversal value for the nverse packng fracton u = Ths value, taken from Sanchez and Cho [4], corresponds to the average value of the rato of the van der Waals densty and the characterstc densty ρ, whch s very chose to the Bond constant 1.3[5]. Wth ths assumpton the lqud-lke volume value, to be used by the mxng rule, s constant for pure polymers. The solvent lqud-lke volume s stll calculated usng the volumetrc thermal expanson coeffcent, as explaned n subsecton Fugacty coeffcents from SCMR The fugacty coeffcent of a substance n a mxture, for any cubc EoS gven by Equaton 1, can be obtaned by [3]: ln ˆφ = b b (Z 1) ln(z β) + q I (14) where b and q are partal molar propertes defned by: ( ) nt k k n wth n T = l n l. T,P,n j (15) For the SCMR mxng rule, the co volume parameter b s gven by the lnear mxng rule, Equaton 5, then b = b. In order to smplfy the notaton n the dervaton of q, let us ntroduce the quantty α for the SCMR mxng rule (Equaton 11): α qi Id = GE ( ) γ RT V b x ln + x V Id q I (16) b whch leads to: q = q + 1 I Id ᾱ α Ī Id I Id (17) and the remanng partal molar propertes are: ( ) V b ᾱ = ln γ ln + V b V Id b V Id b 1 + q I (18) 6

7 ( Ī Id = I Id V + σb + I 0 V Id + σb V ) + ɛb V Id + ɛb (19) where the actvty coeffcent γ should be computed by the chosen Gbbs excess model G E γ. References [1] J. O. Valderrama, Ind. Eng. Chem. Res. 42 (8) (2003) [2] K. Fscher, J. Gmehlng, Flud Phase Equlb. 121 (1-2) (1996) [3] J. M. Smth, H. C. V. Ness, M. M. Abbott, Introducton to Chemcal Engneerng Thermodynamcs, McGraw-Hll, New York, [4] I. C. Sanchez, J. Cho, Polymer 36 (15) (1995) [5] A. Bond, van der Waals Volumes and Rad, J. Phys. Chem. 68 (3) (1964)